Newton Strata in Levi Subgroups

Certain Iwahori double cosets in the loop group of a reductive group, known under the names of $P$-alcoves or $(J,w,\delta)$-alcoves, play an important role in the study of affine Deligne-Lusztig varieties. For such an Iwahori double coset, its Newton stratification is related to the Newton stratification of an Iwahori double coset in a Levi subgroup. We show that this relation is closer than previously known.

Let G be a reductive group over the local field F , whose completion of the maximal unramified extension we denote byF . In the context of Langlands program, one would choose F to be a finite extension of p-adic rationals, whereas in the context of moduli spaces of shtukas, F would be the field of formal Laurent series over a finite field. In any case, the Galois group Γ " GalpF {F q is generated by the Frobenius σ. We moreover pick a σ-stable Iwahori subgroup I Ď GpF q. Then the affine Deligne-Lusztig variety associated to two elements x, b P GpF q is defined as X x pbq " tg P GpF q{I | g´1bσpgq P IxIu.
Evidently, the isomorphism type of X x pbq only depends on the σ-conjugacy class rbs " tg´1bσpgq | g P GpF qu Ď GpF q and the Iwahori double coset IxI. The latter Iwahori double cosets are typically indexed using the extended affine Weyl group Ă W , so that each coset is given by I 9 xI for a uniquely determined element x P Ă W . The set BpGq of σ-conjugacy classes has an important parametrization due to Kottwitz [Kot85;Kot97], characterizing each rbs P BpGq by its Newton point νpbq and Kottwitz point κpbq.
Following [GHN15, Section 2], we assume without loss of generality that the group G is quasi-split over F . We choose a maximal torus T whose unique parahoric subgroup T 0 pF q is contained in I and a σ-stable Borel subgroup T Ă B such that, in the corresponding appartment of the Bruhat-Tits building of GF , the alcove fixed by I is opposite to the dominant cone defined by B. With this notation, the Kottwitz point κpbq for b P GpF q lies in pX˚pT q{ZΦ _ q Γ , where Φ _ is the set of coroots. The dominant Newton point νpbq is an element of X˚pT q Γ 0 b Q, where Γ 0 Ă Γ is the absolute Galois group ofF . We identify the extended affine Weyl group Ă W as the semidirect product of the finite Weyl group W " N G pT q{T with X˚pT q Γ 0 .
Geometric properties of the affine Deligne-Lusztig variety X x pbq are closely related to those of the corresponding Newton stratum rbs X IxI Ă GpF q. Obviously, one is empty if and only if the other is empty. Further properties can be related following [MV20, Section 3.1].
It is an important question to study which of these affine Deligne-Lusztig varieties are non-empty, i.e. to determine the set BpGq x " trbs P BpGq | rbs X IxI ‰ Hu.
An important breakthrough result of Görtz-He-Nie [GHN15] is a characterization of all elements x P Ă W where BpGq x contains the basic σ-conjugacy class. For this purpose, they introduce the notion of a pJ, w, σq-alcove, generalizing the previous notion of a P -alcove known for split groups. Write ∆ Ď Φ for the set of simple roots. Definition 1. Let x P Ă W , w P W and J Ď ∆ such that J " σpJq. Then we say that x is a pJ, w, σq-alcove element if the following conditions are both satisfied: (b) For all positive roots α P Φ`that are not in the root system Φ J generated by J, the corresponding root subgroup U α Ď GpF q satisfies Observe that x is a pJ, w, σq-alcove element if it is a pJ, w 1 , σq-alcove element for any w 1 P wW J , where W J is the subgroup of W generated by the simple reflections coming from J (or equivalently the finite Weyl group of M J ). Following Viehmann [Vie21, Section 4], we say that x is a normalized pJ, w, σq-alcove element if w has minimal length in its coset wW J .
Assume now that x is a normalized pJ, w, σq-alcove element and writex " w´1xσpwq P Ă W M . If rbs P BpGq x , it is a result of Görtz-He-Nie [GHN15, Theorem 3.3.1] that each element in Newton stratum IxI X rbs is of the form i´1wmσpw´1iq for some i P I and m P ď with the union taken over all rb 1 s P BpM q contained in rbs. Our main result states that this union is spurious.
sending a σ-conjugacy class rbs M P BpM qx to the unique σ-conjugacy class rbs G P BpGq with rbs M Ď rbs G . In this case, the dominant Newton points ν M pbq and ν G pbq agree as elements of X˚pT q Γ 0 b Q.
Unfortunately, the relationship discussed here has been a source of confusion in the past. While surjectivity of the map BpM qx Ñ BpGq x follows from [GHN15, Theorem 3.3.1], it should be noted that the natural map BpM q Ñ BpGq is neither injective nor surjective. In particular, for rbs G P BpGq x , the intersection rbs G X M may consist of more than one σ-conjugacy class of M . This point is further discussed in the erratum 1 to [GHN15] and in [Vie21,Example 5.4]. This latter example moreover demonstrates that the assumption of a normalized pJ, w, δq-alcove element is essential for Theorem 2 to hold.
Our theorem together with [GHN15, Theorem 3.3.1] yields a canonical isomorphism Xxpbq Ñ X x pbq, sending gpI X M q P Xxpbq to gw´1I P X x pbq. The geometric correspondence is mirrored by a corresponding representation-theoretic result of He-Nie [HN15, Theorem C], comparing class polynomials of x andx. These are certain structure constants describing the cocenter of the Iwahori-Hecke algebra of Ă W resp. Ă W M . If one knows all class polynomials for a given element x P Ă W , one can use these to determine many geometric properties the affine Deligne-Lusztig varieties X x pbq for rbs P BpGq, cf. [He14, Theorem 6.1]. These properties include dimension as well as the number of top dimensional irreducible components up to the action of the σ-centralizer of b P GpLq. Moreover, in a certain sense, the number of rational points of the Newton stratum IxI X rbs can be expressed using these class polynomials [HNY22, Proposition 3.7]. In this sense, it already follows from [HN15] that these numerical invariants agree for Xxpbq and X x pbq.
Proof of Theorem 2. We follow the Deligne-Lusztig reduction method due to Görtz-He [GH10], i.e. we do an induction on ℓpxq.
If x is of minimal length in its σ-conjugacy class, then BpGq x contains only one element, being the σ-conjugacy class defined by x. Moreover, He-Nie [HN15, Proposition 4.5] prove that in this casex is of minimal length in its σ-conjugacy class in Ă W M . Hence we only have to show that ν M pxq agrees with ν G pxq.
Following the definition of Newton points, we consider σ-twisted powers x σ,n " xσpxq¨¨¨σ n´1 pxq P Ă W .
Observe that each x σ,n is a pJ, w, σ n q-alcove element. Let n be sufficiently large such that x σ,n is a pure translation element, i.e. equal to the image of some µ P X˚pT q Γ 0 in Ă W , and such that σ n is the identity map on Ă W . Then the Newton point ν G pxq is the unique dominant element in the W -orbit of µ{n. Similarly, the Newton point ν M pxq is the unique dominant (with respect to B X M ) element in the W J -orbit of w´1µ{n.
The fact that x σ,n is a pJ, w, 1q-alcove element implies that xw´1µ, αy ě 0 for all α P Φ`zΦ J . Hence ν M pxq is already dominant with respect to B, and thus ν G pxq " ν M pxq. This finishes the proof in case x has minimal length in its σ-conjugacy class.
If x is not of minimal length in its σ-conjugacy class, we can use [HN14, Theorem A] to obtain a sequence for simple affine reflections s i P Ă W and elements x i`1 " s i x i σps i q such that ℓpx 1 q "¨¨¨" ℓpx n q ą ℓpx n`1 q. From [HN15, Lemma 7.1], we find elements w 1 , . . . , w n P W such that each x i is a normalized pJ, w i , σq-alcove element. Denote the corresponding elements bỹ Said proof moreover reveals that eachx i`1 is conjugate tox i either by a simple affine reflection in Ă W M or a length zero element in Ă W M . The Deligne-Lusztig reduction method of Görtz-He [GH10] yields We moreover know from the aforementioned article of He-Nie thatx n`1 "sx n σpsq for some simple affine reflections " w n s n w´1 n P Ă W M of M . Hence BpM qx n " BpM qx n`1 Y BpM qsx n .
We conclude that the map BpM qx Ñ BpGq x is well-defined, surjective and Newton-point preserving. If rb 1 s M , rb 2 s M P BpM qx have the same image rbs G P BpGq x under this map, then ν M pb 1 q " ν G pbq " ν M pb 2 q and κ M pb 1 q " κ M pxq " κ M pb 2 q, hence rb 1 s M " rb 2 s M . This finishes the induction and the proof.
Let us note the following consequence of Theorem 2.
Corollary 3. If x is a pJ, w, σq-alcove element and rb 1 s, rb 2 s P BpGq x , then As an application of this corollary, we prove a conjecture of Dong-Gyu Lim [Lim23], yielding an alternative criterion to the one from [GHN15] for the non-emptiness of the basic Newton stratum in IxI.
Definition 4. Let x P Ă W be written as x " ωs 1¨¨¨sℓpxq for a length zero element ω and simple affine reflections s 1 , . . . , s ℓpxq P Ă W . We define the σ-support of x to be the smallest subset J Ď Ă W containing s 1 , . . . , s ℓpxq and being closed under the action of the composite automorphism σ˝ω. Denote it by supp σ pxq. We say that x is spherically σ-supported if the subgroup of Ă W generated by supp σ pxq is finite.
It follows from [GHN19, Proposition 5.6] that x has spherical σ-support if and only if BpGq x " trbsu for a basic σ-conjugacy class rbs.
Proposition 5. Assume that the Dynkin diagram of Φ is σ-connected, i.e. that the Frobenius σ acts transitively on the irreducible components of the root system Φ.
Let x P Ă W , and denote by rbs P BpGq the unique basic σ-conjugacy class with κpbq " κpxq. Then X x pbq " H if and only if the following two conditions are both satisfied: (a) The element x does not have spherical σ-support, i.e. BpGq x contains a non-basic σ-conjugacy class.
(b) There exists J Ĺ ∆ and w P W such that x is a pJ, w, σq-alcove element.
Proof. If x has spherical σ-support, we get IxI Ď rbs, so that indeed X x pbq ‰ H. In the case that x is not a pJ, w, σq-alcove element for any J Ĺ ∆, we easily obtain X x pbq ‰ H by [GHN15, Theorem A]. Assume now conversely that (a) and (b) both hold true, so we have to show X x pbq " H. Let pJ, wq be as in (b) such that moreover x is a normalized pJ, w, σq-alcove element.
Let rb x s P BpGq x denote the generic σ-conjugacy class.
Assume that rbs P BpGq x . From (b) together with Corollary 3, we see νpbq " νpb x q pmod Φ _ J q.
Since ν G pb x q is dominant and b is basic, we conclude xν G pb x q, αy " 0 for all α P Φ`zΦ J . Thus Φ " Φ J Y Φ J 1 where J 1 Ď ∆ is the stabilizer of νpb x q. By σ-irreducibility and looking at longest roots of irreducible components, we get J " ∆ or J 1 " ∆. We assumed J ‰ ∆ in (b), so we conclude that ν G pb x q must be central. Thus rb x s is basic itself. This contradicts (a).